As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.

We consider nonequilibrium probabilistic dynamics in logisticlike maps $x_{t+1}=1-a|x_t{|}^z$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W\gg 1$ intervals partitioning the phase space and focus on the unique value $q_{\mathrm{s}\mathrm{e}\mathrm{n}}<1$ for which the entropic form $S_q\equiv (1-\sum _{i=1}^W{p}_i^q)/(q-1)$ linearly increases with time. We then verify that $S_{q_{\mathrm{s}\mathrm{e}\mathrm{n}}}(t)-S_{q_{\mathrm{s}\mathrm{e}\mathrm{n}}}(\infty )$ vanishes like $t^{-1/[q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)-1]}$ [$q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)>1$]. We finally exhibit a new finite-size scaling, $q_{\mathrm{r}\mathrm{e}\mathrm{l}}(\infty )-q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)\propto W^{-|q_{\mathrm{s}\mathrm{e}\mathrm{n}}|}$. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.